Status of the Beam-Based Feedback for the CLIC main linac

CERN, BE-ABP (Accelerators and Beam Physics group)

Jürgen Pfingstner Status of the Beam-Based Feedback for the CLIC main linac

Status of the Beam-Based Feedbackfor the CLIC main linac

Jürgen Pfingstner

14th of October 2009



Content

1. Review of the work on the BBF

2. Idea of an adaptive controller

3. Problems with an adaptive scheme and possible solutions

ControllerAccelerator

R

System

identification

Controller

design

Param.

riui+1

yi

Ri & gmi

Adaptive Controller (STR) System



The model of the main linac

1.) Perfect aligned beam line

BPMiQPi

Laser-straight

beam

2.) One misaligned QP

yixi

Betatron- oscillation

(given by the beta function of

the lattice)yjxj(=0)

a.) 2 times xi -> 2 times amplitude

-> 2 times yj

b.) xi and xj are independent

Þ Linear system without ‘memory’

2

1

2221

1211

2

1

xx

rrrr

yy

Rxy Þ

y … vector of BPM readings

x … vector of the QP displacements

R … response matrix



The matrix R• N’s Columns correspond to the measured

beam motion in the linac, created by the N’s QP

• Motion is characterized by phase advance , beta function and Landau damping

• R is ‘nearly’ triangular and the elemets close to the diagonal are most important.

-1

-1-1

…

0



Robustness study and properties of the system

System properties summarized:• Main linac is a discrete memoryless

FIFO system (simple), but MIMO

• The system in in principal easy to control, since there is no inner dynamic (dynamic just by feedback).

Character of the control problem: • Classical feedback design objectives

as stability and set point following are not important issues,

• Minimal steady state error, due to noise and disturbances and very good system knowledge matters.

• Focus is more on precision and not on robustness

Robustness study [1]:

• Feedback with nominal R applied to not-nominal accelerator

• Simulations of the feedback performance in PLACET [2]

• Feedback was the dead-beat controller (see next slide)

Results:

• Outcome was a table of still valid accelerator parameters

• Message:

• System is by itself very robust against imperfect system

knowledge

• Stability is not an big issue



State controller• Idea: Calculate the QP

positions of the last step and correct them [3].

• Corresponds to a state controller [4], that puts all poles to zero (deadbeat contr.)

•z

1

• Set point transfer function:

• Deadbeat controller [5]:

- Very fast ground motion rejection and

set point following

- but introduces a lot of noise from the

BPMs in the system

Alternative state controller:

• Apply not full correction

but

• Factor g balances between speed and noise, by moving

to poles further away from zero



Emittance based controller [6]• Idea: Emittance as a function of

normalized beam macro particle coordinates at the end of the linac

• Optimizing feedback for min. emittance growth and min. BPM offset (quadratic sense)

• Result is a 10 times smaller growth rate.

• Design uses SVD decomposition but is not a SVD controller (no diagonalization).

• Problem: Controller design uses macro particle

coordinates that cannot be measured in reality.

• Controller has to rely on simulated data.

• Practical usefulness is questionable and has to be verified,

by robust performance evaluation.



Idea of an adaptive controller

ControllerAccelerator

R

System

identification

Controller

design

Param.

riui+1

yi

Ri & gmi

Adaptive Controller (STR) System

• 3 adaptive control schema [7]:

- Model-Reference Adapt. Sys. (MRAS)

- Self-tuning Regulators (STR)

- Dual Control

STR

• Previous designs do not take into account system changes.

• Idea: Tackle problem of system changes by an online system

identification

• Lear about the system by:

- Input data

- Output data

- Guess about the system structure

• Usage:

- For system diagnostics and input for different feedbacks (keep R as

it was)

- Input for an online controller design



System identification• Real system:

• Model system:

• Goal:

Fit the model system in some

sense to the real system,

using and

... Input data

… Output data

… Ground motion

… white and gaussian noise (always here)



RLS algorithm and derivative• can e.g. be

formalized as

• Offline solution to this Least Square problem by pseudo inverse (Gauss):

... Estimated parameter

… Input data

… Output data

• LS calculation can be modified for recursive

calculation (RLS):

• is a forgetting factor for time varying systems

• Derivatives (easier to calculate)

- Projection algorithm (PA)

- Stochastic approximation (SA)

- Least Mean Square (LMS)



Computational effort• Normally the computational

effort for RLS is very high.

• For most general form of linac problem size:- Matrix inversion (1005x1005)

- Storage of matrix P (1 TByte)

• Therefore often just simplifications as PA, SA and LMS are used.

• For the linac system and

have a simple diagonal form.

• The computational effort can be reduced strongly

- Matrix inversion becomes scalar inversion

- P (few kByte)

- Parallelization is possible

• Full RLS can be calculated easily



• Noise/Drift generation

• Parameter of noise (for similar emittance growth; ΔT = 5s):- BPM noise: white noise (k = 5x10-8)- RF disturbance: 1/f2 drift (k = 7x10-4) + white noise (k = 1.5x10-2)- QP gradient errors: 1/f2 drift (k = 4x10-6) + white noise (k = 3x10-4)- Ground motion: According to Model A of A. Sery [8]

• RF drift much more visible in parameter changes than QP errors

Modeling of the system change

Random number z-1

+z-1

+k

white noise 1/f

noise 1/f2

noisewhite and gaussian



First simulation results

• Identification of one line of R and the gm-vector d • Simulation data from PLACET

• ΔT = 5s

• λ = 0.85

• R changes according to

last slide

• Ground motion as by A.

Seri (model A [8])



Forgetting factor λ

d10: λ to big (overreacting)d500: λ fitsd1000: λ is to small

=> Different positions in the linac should use a different λ (work)



Problems with the basic approach

Problem 2: Nature of changes

• No systematic in system change• Adding up of many indep. Changes• Occurs after long excitation

Problem 1: Excitation

• Particles with different energies move

differently

• If beam is excited, these different movements

lead to filamentation in the phase space

(Landau Damping)

• This increases the emittance

=> Excitation cannot be arbitrary



Semi-analytic identification scheme

Δx’1 Δ x’2 Δ x’3

Excitation Strategy:

• Necessary excitation can not be arbitrary, due to

emittance increase

• Strategy: beam is just excited over short distance and

caught again.

• Beam Bump with min. 3 kickers is necessary

Practical system identification:

• Just parts of R can be identified

• Rest has to be interpolated

- Transient landau damping model

- Algorithm to calculate phase advance from BPM/R data

-1

-1-1

…0



Model of the transient Landau DampingResult: (Kick at 390 and

6350m)• Approach [9]:

• Envelope by peak detection algorithm

• Limitation: Works just for time independent

energy distribution

• Not the case at injection into linac => fit to data



Open questions• Strategy of determine in an way, that the knowledge about the

disturbance signals is best possible used.• Gaining knowledge of the best possible excitation of the beam

without loosing to much beam quality.• Getting more detailed information about the nature of many

disturbances to tailor the algorithm accordingly (not only RLS is possible).

Resume• The approach of an adaptive controller is in principle good, but

• There are many accumulating inaccuracies as:

- Landau Model

- Phase advance reconstruction

- Remaining Jitter in the estimated model

- Undeterministic propagation of disturbances

• Hopefully these inaccuracies do not destroy the practical usability!!!



References[1] J. Pfingstner, W. http://indico.cern.ch/conferenceDisplay.py?confId=54934. Beam-based

feedback for the main linac, CLIC Stabilisation Meeting 5, 30th March 2009.[2] E. T. dAmico, G. Guignard, N. Leros, and D. Schulte. Simulation Package based on

PLACET. In Proceedings of the 2001 Particle Accelerator Converence (PAC01), volume 1, pages 3033–3035, 2001.

[3] A. Latina and R. Tomas G. Rumolo, D. Schulte. Feedback studies. Technical report, EUROTeV, 2007. EUROTeV Report 2007 065.

[4] Otto Föllinger. Einführung in die Methoden und ihre Anwendung. Hüthig Buch Verlag Heidelberg, 1994. ISBN: 3-7785-2915-3.

[5] Nicolaos Dourdoumas and Martin Horn. Regelungstechnik. Pearson Studium, 2003. ISBN: 3-8273-7059-0.

[6] Peder Eliasson. Dynamic imperfections and optimized feedback design in the compact linear collider main linac. Phys. Rev. Spec. Top. Accel. Beams, 11:51003, 2008.

[7] K. J. Åström and B. Wittenmark. Adaptive Control. Dover Publications, Inc., 2008. ISBN: 0-486-46278-1.

[8] Andrey Sery and Olivier Napoly. Influence of ground motion on the time evolution of beams in linear colliders. Phys. Rev. E, 53:5323, 1996.

[9] Alexander W. Chao. Physics of Collective Beam Instabilities in High Energy Accelerators. John Wiley & Sons, Inc., 1993. ISBN: 0-471-55184-8.



Thank you for your attention!

Status of the Beam-Based Feedback for the CLIC main linac

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